A curve of class () is locally convex if the vectors are a positive basis to for all . Given an integer and , let be the set of all locally convex curves with fixed initial and final Frenet frame and . Saldanha and Shapiro proved that there are just finitely many non-homeomorphic spaces among when varies in (in particular, at most for ). For any , one of these spaces is proved to be homeomorphic to the (well understood) space of generic curves (see below), but very little is known in general about the others. For , Saldanha determined the homotopy type of the spaces . The purpose of this work is to study the case . We will obtain information on the homotopy type of one of these two other spaces, allowing us to conclude that none of the connected components of is homeomorphic to a connected component of the space of generic curves.
La courbe de classe () est localement convexe si les vecteurs forment une base positive de pour chaque . Pour un entier et , soit l’ensemble de toutes les courbes localement convexes avec repères de Frenet initial et final fixes et . Saldanha et Shapiro ont demontré qu’il n’y a qu’un nombre fini d’espaces non-homéomorphes parmi les avec (en particulier, au plus pour ). Pour , ils demontrent qu’un de ces espaces est homéomorphe à l’espace (bien compris) des courbes génériques (défini ci-dessous) mais on connaît très peu les autres espaces. Pour , Saldanha a déterminé le type d’homotopie des espaces . Le but de ce travail est d’étudier le cas . On obtient des informations sur le type d’homotopie d’un de ces autres deux espaces, ce qui nous permet de déduire qu’aucune des composantes connexes de n’est homéomorphe à une composante connexe de l’espace des courbes génériques.
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Accepted:
Published online:
DOI: 10.5802/aif.3267
Keywords: Locally convex curves, homotopy type, Bruhat decomposition
Mot clés : Courbes localement convexes, type d’homotopie, décomposition de Bruhat
Alves, Emília 1; Saldanha, Nicolau C. 1
@article{AIF_2019__69_3_1147_0, author = {Alves, Em{\'\i}lia and Saldanha, Nicolau C.}, title = {Results on the homotopy type of the spaces of locally convex curves on $\protect \mathbb{S}^3$}, journal = {Annales de l'Institut Fourier}, pages = {1147--1185}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {69}, number = {3}, year = {2019}, doi = {10.5802/aif.3267}, zbl = {07067428}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3267/} }
TY - JOUR AU - Alves, Emília AU - Saldanha, Nicolau C. TI - Results on the homotopy type of the spaces of locally convex curves on $\protect \mathbb{S}^3$ JO - Annales de l'Institut Fourier PY - 2019 SP - 1147 EP - 1185 VL - 69 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3267/ DO - 10.5802/aif.3267 LA - en ID - AIF_2019__69_3_1147_0 ER -
%0 Journal Article %A Alves, Emília %A Saldanha, Nicolau C. %T Results on the homotopy type of the spaces of locally convex curves on $\protect \mathbb{S}^3$ %J Annales de l'Institut Fourier %D 2019 %P 1147-1185 %V 69 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3267/ %R 10.5802/aif.3267 %G en %F AIF_2019__69_3_1147_0
Alves, Emília; Saldanha, Nicolau C. Results on the homotopy type of the spaces of locally convex curves on $\protect \mathbb{S}^3$. Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 1147-1185. doi : 10.5802/aif.3267. https://aif.centre-mersenne.org/articles/10.5802/aif.3267/
[1] Topology of the spaces of locally convex curves on the 3-sphere, Ph. D. Thesis, PUC-Rio, Rio de Janeiro (Brazil) (2016)
[2] Convex curves in , Proc. Steklov Inst. Math., Volume 221 (1998) no. 2, pp. 3-39 | Zbl
[3] The geometry of spherical curves and the algebra of quaternions, Russ. Math. Surv., Volume 50 (1995) no. 1, pp. 1-68 | Zbl
[4] Results on infinite dimensional topology and applications to the structure of the critical sets of nonlinear Sturm–Liouville operators, J. Differ. Equations, Volume 188 (2003) no. 2, pp. 569-590 | DOI | MR | Zbl
[5] Existence of -principle for Engel structures (2015) (https://arxiv.org/abs/1507.05342) | Zbl
[6] On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents, Am. J. Math., Volume 79 (1957) no. 3, pp. 497-516 | DOI | MR | Zbl
[7] Introduction to the h-principle, Graduate Studies in Mathematics, 48, American Mathematical Society, 2002, xvii+206 pages | MR | Zbl
[8] Ãber Krümmung und Windung geschlossener Raumkurven, Math. Ann., Volume 101 (1929), pp. 238-252 | DOI | MR | Zbl
[9] Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 9, Springer, 1986, ix+363 pages | MR | Zbl
[10] Immersions of manifolds, Trans. Am. Math. Soc., Volume 93 (1959), pp. 242-276 | DOI | MR | Zbl
[11] Nondegenerate curves on S and orbit classification of the Zamolodchikov algebra, Commun. Math. Phys., Volume 145 (1992) no. 2, pp. 357-362 | MR | Zbl
[12] Nondegenerate homotopies of curves on the unit 2-sphere, J. Differ. Geom., Volume 4 (1970) no. 3, pp. 339-348 | DOI | MR | Zbl
[13] Flexibility for tangent and transverse immersions in Engel manifolds (2016) (https://arxiv.org/abs/1609.09306) | Zbl
[14] Optimal paths for a car that goes both forwards and backwards, Pac. J. Math., Volume 145 (1990) no. 2, pp. 367-393 | DOI | MR
[15] The homotopy and cohomology of spaces of locally convex curves in the sphere - I (2009) (https://arxiv.org/abs/0905.2111v1)
[16] The homotopy and cohomology of spaces of locally convex curves in the sphere - II (2009) (https://arxiv.org/abs/0905.2116v1)
[17] The homotopy type of spaces of locally convex curves in the sphere, Geom. Topol., Volume 19 (2015), pp. 1155-1203 | DOI | MR | Zbl
[18] Spaces of locally convex curves in and combinatorics of the group , J. Singul., Volume 4 (2012), pp. 1-22 | MR | Zbl
[19] Homotopy classification of nondegenerate quasiperiodic curves on the 2-sphere, Publ. Inst. Math., Nouv. Sér., Volume 66 (1999) no. 80, pp. 127-156 | MR | Zbl
[20] On the number of connected components in the space of closed nondegenerate curves on , Bull. Am. Math. Soc., Volume 25 (1991) no. 1, pp. 75-79 | DOI | MR | Zbl
[21] Topology of the space of nondegenerate curves, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 57 (1993) no. 5, pp. 106-126 | Zbl
[22] The classification of immersions of spheres in Euclidean spaces, Ann. Math., Volume 69 (1959) no. 2, pp. 327-344 | MR | Zbl
[23] Homotopies of curves on the 2-sphere with geodesic curvature in a prescribed interval, Ph. D. Thesis, PUC-Rio, Rio de Janeiro (Brazil) (2013) | Zbl
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